Mean-Gini,Portfolio Theory,and the Pricing of Risky Assets

This paper presents the mean-Gini (MG) approach to analyze risky prospects and construct optimum portfolios. The proposed method has the simplicity of a mean-variance model and the main features of stochastic dominance efficiency. Since mean-Gini is consistent with investor behavior under uncertainty for a wide class of probability distributions, Gini's mean difference is shown to be more adequate than the variance for evaluating the variability of a prospect. The MG approach is then applied to capital markets and the security valuation theorem is derived as a general relationship between average return and risk. This is further extended to include a degree of risk aversion that can be estimated from capital market data. The analysis is concluded with the concentration ratio to allow for the classification of different securities with respect to their relative riskiness.     

THE MEAN‐GINI EFFICIENT PORTFOLIO FRONTIER

A main advantage of the mean‐variance (MV) portfolio frontier is its simplicity and ease of derivation. A major shortcoming, however, lies in its familiar restrictions, such as the quadraticity of preferences or the normality of distributions. As a workable alternative to MV, we present the mean‐Gini (MG) efficient portfolio frontier. Using an optimization algorithm, we compute MG and mean‐extended Gini (MEG) efficient frontiers and compare the results with the MV frontier. MEG allows for the explicit introduction of risk aversion in building the efficient frontier. For U.S. classes of assets, MG and MEG efficient portfolios constructed using Ibbotson (2000) monthly returns appear to be more diversified than MV portfolios. When short sales are allowed, distinct investor risk aversions lead to different patterns of portfolio diversification, a result that is less obvious when short sales are foreclosed. Furthermore, we derive analytically the MG efficient portfolio frontier by restricting asset distributions. The MG frontier derivation is identical in structure to that of the MV efficient frontier derivation. The penalty paid for simplifying the search for the MG efficient frontier is the loss of some information about the distribution of assets.     

Portfolio selection based on the mean–VaR efficient frontier

Value-at-Risk (VaR) has become one of the standard measures for assessing risk not only in the financial industry but also for asset allocations of individual investors. The traditional mean–variance framework for portfolio selection should, however, be revised when the investor's concern is the VaR instead of the standard deviation. This is especially true when asset returns are not normal. In this paper, we incorporate VaR in portfolio selection, and we propose a mean–VaR efficient frontier. Due to the two-objective optimization problem that is associated with the mean–VaR framework, an evolutionary multi-objective approach is required to construct the mean–VaR efficient frontier. Specifically, we consider the elitist non-dominated sorting Genetic Algorithm (NSGA-II). From our empirical analysis, we conclude that the risk-averse investor might inefficiently allocate his/her wealth if his/her decision is based on the mean–variance framework.     

The econometrics of efficient portfolios

It is well known that the standard mean variance approach can be inappropriate when return distributions feature skewness, fat tails or multimodes. This is typically the situation for portfolios including derivatives. In this case, it can be necessary to come back to the basic expected utility approach. In this paper, an efficient portfolio maximizes the expected utility of future wealth. This paper presents an analysis of the efficiency frontier, formed by a set of efficient portfolios corresponding to a parameterized class of utility functions. First, we discuss the estimation of an efficient portfolio and introduce several tests of the efficiency hypothesis, depending on what is known about the utility function and the budget level. Next we analyse the shape of the frontier and develop a procedure for testing the separability of the efficiency frontier into K independent funds. The inference is semi-nonparametric because the return distribution is left unspecified. We illustrate our approach by an application to portfolios including derivatives.     

Short Sales Restrictions and Kinks on the Mean Variance Frontier

With a short sales restriction, there may be switching points along the mean variance frontier corresponding to changes in the set of assets held. Traditional wisdom holds that each switching point corresponds to a kink, while Ross has claimed that kinks never occur. This paper shows that the truth lies between the two views, since the efficient frontier may or may not be kinked at a switching point. There is some indication that kinks are rare, since a kink corresponds to a portfolio in which all assets have the same expected return.     

MEAN GINI CAPITAL ASSET PRICING MODEL: SOME EMPIRICAL EVIDENCE

The capital asset pricing model of Sharpe (1964) and Lintner (1965) provides a valid approach to portfolio selection if either the distribution of asset returns is jointly normal or the investor's preference function is quadratic. Various authors have questioned the validity of these assumptions, and Roll (1977) raises the question whether the traditional CAPM can be tested. An alternative Capital Asset Pricing Model has been proposed by Shalit and Yitzhaki (1984). In this model the extended mean Gini coefficient is used to measure risk. As little research has been conducted on this model, this paper estimates systematic risk as derived from the extended mean Gini model for a sample of Australian companies and compares the empirical security market line with the predicted extended mean Gini security market line.     

The premium of dynamic trading

It is well established that, in a market with inclusion of a risk-free asset, the single-period mean–variance efficient frontier is a straight line tangent to the risky region, a fact that is the very foundation of the classical CAPM. In this paper, it is shown that, in a continuous-time market where the risky prices are described by Itô processes and the investment opportunity set is deterministic (albeit time-varying), any efficient portfolio must involve allocation to the risk-free asset at any time. As a result, the dynamic mean–variance efficient frontier, although still a straight line, is strictly above the entire risky region. This in turn suggests a positive premium, in terms of the Sharpe ratio of the efficient frontier, arising from dynamic trading. Another implication is that the inclusion of a risk-free asset boosts the Sharpe ratio of the efficient frontier, which again contrasts sharply with the single-period case.     

Dynamic asset–liability management in a Markov market with stochastic cash flows

This paper provides a general model to investigate an asset–liability management (ALM) problem in a Markov regime-switching market in a multi-period mean–variance (M–V) framework. Emphasis is placed on the stochastic cash flows in both wealth and liability dynamic processes, and the optimal investment and liquidity management strategies in achieving the M–V bi-objective of terminal surplus are evaluated. In this model, not only the asset returns and liability returns, but also the cash flows depend on the stochastic market states, which are assumed to follow a discrete-time Markov chain. Adopting the dynamic programming approach, the matrix theory and the Lagrange dual principle, we obtain closed-form expressions for the efficient investment strategy. Our proposed model is examined through empirical studies of a defined contribution pension fund. In-sample results show that, given the same risk level, an ALM investor (a) starting in a bear market can expect a higher return compared to beginning in a bull market and (b) has a lower expected return when there are major cash flow problems. The effects of the investment horizon and state-switching probability on the efficient frontier are also discussed. Out-of-sample analyses show the dynamic optimal liquidity management process. An ALM investor using our model can achieve his or her surplus objective in advance and with a minimum variance close to zero.     

The Relationship between Bankruptcy Model Predictions and Stock Market Perceptions of Bankruptcy

This study uses a cumulative sum technique to determine the point at which the stock market first perceives that a firm may file for bankruptcy. The study then attempts to identify information, whether from financial statements or from other sources, that may have influenced the market in its reassessment of the firm's prospects. The results indicate that the switching point of the mean and variance of stock returns appears to be related both to financial statement information (as measured by changes in bankruptcy model probability assessments) and the release of unfavorable news in the Wall Street Journal.     

Capital market equilibrium with heterogeneous investors

As a two-parameter model that satisfies stochastic dominance, the mean-extended Gini model is used to build efficient portfolios. The model quantifies risk aversion heterogeneity in capital markets. In a simple Edgeworth box framework, we show how capital market equilibrium is achieved for risky assets. This approach provides a richer basis for analysing the pricing of risky assets under heterogeneous preferences. Our main results are: (1) identical investors, who use the same statistic to represent risk, hold identical portfolios of risky assets equal to the market portfolio; and (2) heterogeneous investors as expressed by the variance or the extended Gini hold different risky assets in portfolios, and therefore no one holds the market portfolio.     

Do environmental variables affect the performance and technical efficiency of the European banking systems? A parametric analysis using the stochastic frontier approach

The paper compares the efficiency of the European banking systems in view of the constitution of the European Monetary Union. Since competition among banks will increase, it is important to identify the most efficient banking system able to play a role in that market. A parametric approach is adopted, based on the estimation of a stochastic cost frontier. This methodology enables one to measure X-inefficiency and to model it as a function of environmental variables which may influence firms' efficiency. By means of this analysis it is possible to identify the most efficient banking systems and to focus on the determinants of deviations from cost minimizing. The analysis highlights significant efficiency gaps among the performances of banks in different countries and of different institutional types. In particular, it is found that the Mittel-European model is the one that operates closest to the efficient frontier. This may indicate that, compared with separated banks, the universal banking system allows for production plans which come closer to the optimal frontier. The analysis suggests that, at the beginning of European Monetary Union, national barriers and regulatory frameworks are still responsible for deviation from the efficient frontiers.     

How does beta explain stochastic dominance efficiency?

Stochastic dominance rules provide necessary and sufficient conditions for characterizing efficient portfolios that suit all expected utility maximizers. For the finance practitioner, though, these conditions are not easy to apply or interpret. Portfolio selection models like the mean–variance model offer intuitive investment rules that are easy to understand, as they are based on parameters of risk and return. We present stochastic dominance rules for portfolio choices that can be interpreted in terms of simple financial concepts of systematic risk and mean return. Stochastic dominance is expressed in terms of Lorenz curves, and systematic risk is expressed in terms of Gini. To accommodate for risk aversion differentials across investors, we expand the conditions using the extended Gini.     

An analysis of portfolio selection with background risk

This paper investigates the impact of background risk on an investor’s portfolio choice in a mean–variance framework, and analyzes the properties of efficient portfolios as well as the investor’s hedging behaviour in the presence of background risk. Our model implies that the efficient portfolio with background risk can be separated into two independent components: the traditional mean–variance efficient portfolio, and a self-financing component constructed to hedge against background risk. Our analysis also shows that the presence of background risk shifts the efficient frontier of financial assets to the right with no changes in its shape. Moreover, both the composition of the hedge portfolio and the location of the efficient frontier are greatly affected by a number of background risk factors, including the proportion of background assets in total wealth and the correlation between background risk and financial risk.     

Risky asset pricing based on safety first fund management

We study a portfolio selection model based on Kataoka's safety-first criterion (KSF model in short). We assume that the market is complete but without risk-free asset, and that the returns are jointly elliptically distributed. With these assumptions, we provide an explicit analytical optimal solution for the KSF model and obtain some geometrical properties of the efficient frontier in the plane of probability risk degree z _α and target return r _α. We further prove a two-fund separation and tangency portfolio theorem in the spirit of the traditional mean-variance analysis. We also establish a risky asset pricing model based on risky funds that is similar to Black's zero-beta capital asset pricing model (CAPM, for short). Moreover, we simplify our risky asset pricing model using a derivative risky fund as a reference for market evaluation.     

Multi-scale variation,path risk and long-term portfolio management

Mean–variance analysis is constrained to weight the frequency bands in a return time series equally. A more flexible approach allows the user to assign preference weightings to short or longer run frequencies. Wavelet analysis provides further flexibility, removing the need to assume asset returns are stationary and encompassing alternative return concepts. The resulting portfolio choice methodology establishes a reward–energy efficient frontier that allows the user to trade off expected reward against path risk, reflecting preferences as between long or short run variation. The approach leads to dynamic analogues of mean–variance such as band pass portfolios that are more sensitive to variability at designated scales.     

Optimal rules for ordering uncertain prospects

In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of alternatives, those that are inferior (for each utility function in the restricted class) to a member of the given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule when comparing uncertain prospects with equal means. We also show that in the general case of unequal means, no known selection rule uses both necessary and sufficient conditions for dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest admissible set. The TSD rule is complex and we provide an efficient algorithm to obtain the TSD admissible set. For certain restrictive classes of the probability distributions (of returns on uncertain prospects) which cover most commonly used distributions in finance and economics, we obtain the optimal rule and show that it reduces to a simple form. We also study the relationship of the optimal selection rule to others previously advocated in the literature, including the more popular mean-variance rule as well as the semi-variance rule.     

学前教育经费投入水平的地区差异与成因——基于省级数据的实证分析

利用2002~2011年31个省、市、自治区的面板数据测算我国学前教育生均经费地区差异的基尼系数,并对其进行静态和动态分解。结果表明,学前教育三年行动计划的实施对促进学前教育的公平发展起到了一定的作用,但地区差异仍十分突显。根据地区学前教育投入水平差异的形成原因,应加大政府投入,强化教育公平,支持民间投资,完善贫困资助制度。     

基于跳跃扩散过程的保险资金最优投资模型研究

保险公司的盈余为跳跃扩散过程,保险人投资于债券和股票,且股票的价格服从跳跃扩散过程的最优投资组合。在均值-方差准则下通过随机最优控制方法,建立并求解保险资金投资模型的HJB方程,获得了保险资金最优投资模型和有效边界的闭式解,并进行了数值模拟。结果显示,投资于风险证券的资金量与初始资本金并不是简单的正比例关系。     

Linear beta pricing with efficient/inefficient benchmarks and short-selling restrictions

We provide two security pricing models that can be used when short sales of risky securities are not permitted. The first model uses a benchmark located on the expected return-standard deviation efficient frontier without short sales and presents security expected returns as a weighted linear function of two betas, one induced by the benchmark and the other adjusting for the short-sale constraints. The second model uses a benchmark that is inefficient relative to the efficient frontier, does not allow short sales, and expresses security returns as a weighted linear function of three betas:one induced by the inefficient benchmark, and the other two adjusting simultaneously for the short-sales restrictions and the benchmark's inefficiency.These results complement the linear pricing models that link expected returns and betas by allowing, for efficient or inefficient expected return-standard deviation, benchmarks with restricted short sales.     

Dynamic trading with uncertain exit time and transaction costs in a general Markov market

This paper investigates a dynamic trading problem with transaction cost and uncertain exit time in a general Markov market, where the mean vector and covariance matrix of returns depend on the states of the stochastic market, and the market state is regime switching in a time varying state set. Following the framework proposed by Gârleanu and Pedersen (2013), the investor maximizes his or her multi-period mean–variance utility, net of quadratic transaction costs capturing the linear price impact where trades lead to temporary linear changes in prices. The explicit expression for the optimal strategy is derived by using matrix theory technique and dynamic programming approach. Finally, numerical examples are provided to study the effects of transition cost and exit probability on the wealth process, the trading strategy, turnover rate and the total transaction cost.